7.12 Representable sheaves
Let $\mathcal{C}$ be a category. The canonical topology is the finest topology such that all representable presheaves are sheaves (it is formally defined in Definition 7.47.12 but we will not need this). This topology is not always the topology associated to the structure of a site on $\mathcal{C}$. We will give a collection of coverings that generates this topology in case $\mathcal{C}$ has fibered products. First we give the following general definition.
Definition 7.12.1. Let $\mathcal{C}$ be a category. We say that a family $\{ U_ i \to U\} _{i \in I}$ is an effective epimorphism if all the morphisms $U_ i \to U$ are representable (see Categories, Definition 4.6.4), and for any $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the sequence
\[ \xymatrix{ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, X) \ar[r] & \prod \nolimits _{i \in I} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_ i, X) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \nolimits _{(i, j) \in I^2} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_ i \times _ U U_ j, X) } \]
is an equalizer diagram. We say that a family $\{ U_ i \to U\} $ is a universal effective epimorphism if for any morphism $V \to U$ the base change $\{ U_ i \times _ U V \to V\} $ is an effective epimorphism.
The class of families which are universal effective epimorphisms satisfies the axioms of Definition 7.6.2. If $\mathcal{C}$ has fibre products, then the associated topology is the canonical topology. (In this case, to get a site argue as in Sets, Lemma 3.11.1.)
Conversely, suppose that $\mathcal{C}$ is a site such that all representable presheaves are sheaves. Then clearly, all coverings are universal effective epimorphisms. Thus the following definition is the “correct” one in the setting of sites.
Definition 7.12.2. We say that the topology on a site $\mathcal{C}$ is weaker than the canonical topology, or that the topology is subcanonical if all the coverings of $\mathcal{C}$ are universal effective epimorphisms.
A representable sheaf is a representable presheaf which is also a sheaf. Since it is perhaps better to avoid this terminology when the topology is not subcanonical, we only define it formally in that case.
Definition 7.12.3. Let $\mathcal{C}$ be a site whose topology is subcanonical. The Yoneda embedding $h$ (see Categories, Section 4.3) presents $\mathcal{C}$ as a full subcategory of the category of sheaves of $\mathcal{C}$. In this case we call sheaves of the form $h_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ representable sheaves on $\mathcal{C}$. Notation: Sometimes, the representable sheaf $h_ U$ associated to $U$ is denoted $\underline{U}$.
Note that we have in the situation of the definition
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U, \mathcal{F}) = \mathcal{F}(U) \]
for every sheaf $\mathcal{F}$, since it holds for presheaves, see (7.2.1.1). In general the presheaves $h_ U$ are not sheaves and to get a sheaf you have to sheafify them. In this case we still have
7.12.3.1
\begin{equation} \label{sites-equation-map-representable-into-sheaf} \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) = \mathcal{F}(U) \end{equation}
for every sheaf $\mathcal{F}$. Namely, the first equality holds by the adjointness property of $\# $ and the second is (7.2.1.1).
slogan
Lemma 7.12.4. Let $\mathcal{C}$ be a site. If $\{ U_ i \to U\} _{i \in I}$ is a covering of the site $\mathcal{C}$, then the morphism of presheaves of sets
\[ \coprod \nolimits _{i \in I} h_{U_ i} \to h_ U \]
becomes surjective after sheafification.
Proof.
By Lemma 7.11.2 above we have to show that $\coprod \nolimits _{i \in I} h_{U_ i}^\# \to h_ U^\# $ is an epimorphism. Let $\mathcal{F}$ be a sheaf of sets. A morphism $h_ U^\# \to \mathcal{F}$ corresponds to a section $s \in \mathcal{F}(U)$. Hence the injectivity of $\mathop{\mathrm{Mor}}\nolimits (h_ U^\# , \mathcal{F}) \to \prod _ i \mathop{\mathrm{Mor}}\nolimits (h_{U_ i}^\# , \mathcal{F})$ follows directly from the sheaf property of $\mathcal{F}$.
$\square$
The next lemma says, in the case the topology is weaker than the canonical topology, that every sheaf is made up out of representable sheaves in a way.
Lemma 7.12.5. Let $\mathcal{C}$ be a site. Let $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset such that every object of $\mathcal{C}$ has a covering by elements of $E$. Let $\mathcal{F}$ be a sheaf of sets. There exists a diagram of sheaves of sets
\[ \xymatrix{ \mathcal{F}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_0 \ar[r] & \mathcal{F} } \]
which represents $\mathcal{F}$ as a coequalizer, such that $\mathcal{F}_ i$, $i = 0, 1$ are coproducts of sheaves of the form $h_ U^\# $ with $U \in E$.
Proof.
First we show there is an epimorphism $\mathcal{F}_0 \to \mathcal{F}$ of the desired type. Namely, just take
\[ \mathcal{F}_0 = \coprod \nolimits _{U \in E, s \in \mathcal{F}(U)} (h_ U)^\# \longrightarrow \mathcal{F} \]
Here the arrow restricted to the component corresponding to $(U, s)$ maps the element $\text{id}_ U \in h_ U^\# (U)$ to the section $s \in \mathcal{F}(U)$. This is an epimorphism according to Lemma 7.11.2 and our condition on $E$. To construct $\mathcal{F}_1$ first set $\mathcal{G} = \mathcal{F}_0 \times _\mathcal {F} \mathcal{F}_0$ and then construct an epimorphism $\mathcal{F}_1 \to \mathcal{G}$ as above. See Lemma 7.11.3.
$\square$
Lemma 7.12.6. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets on $\mathcal{C}$. Then there exists a diagram $\mathcal{I} \to \mathcal{C}$, $i \mapsto U_ i$ such that
\[ \mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}} h_{U_ i}^\# \]
Moreover, if $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is a subset such that every object of $\mathcal{C}$ has a covering by elements of $E$, then we may assume $U_ i$ is an element of $E$ for all $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$.
Proof.
Let $\mathcal{I}$ be the category whose objects are pairs $(U, s)$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $s \in \mathcal{F}(U)$ and whose morphisms $(U, s) \to (U', s')$ are morphisms $f : U \to U'$ in $\mathcal{C}$ with $f^*s' = s$. For each object $(U, s)$ of $\mathcal{I}$ the element $s$ defines by the Yoneda lemma a map $s : h_ U \to \mathcal{F}$ of presheaves. Hence by the universal property of sheafification a map $h_ U^\# \to \mathcal{F}$. These maps are immediately seen to be compatible with the morphisms in the category $\mathcal{I}$. Hence we obtain a map $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U \to \mathcal{F}$ of presheaves (where the colimit is taken in the category of presheaves) and a map $\mathop{\mathrm{colim}}\nolimits _{(U, s)} (h_ U)^\# \to \mathcal{F}$ of sheaves (where the colimit is taken in the category of sheaves). Since sheafification is the left adjoint to the inclusion functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{PSh}(\mathcal{C})$ (Proposition 7.10.12) we have $\mathop{\mathrm{colim}}\nolimits (h_ U)^\# = (\mathop{\mathrm{colim}}\nolimits h_ U)^\# $ by Categories, Lemma 4.24.5. Thus it suffices to show that $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U \to \mathcal{F}$ is an isomorphism of presheaves. To see this we show that for every object $X$ of $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U(X) \to \mathcal{F}(X)$ is bijective. Namely, it has an inverse sending the element $t \in \mathcal{F}(X)$ to the element of the set $\mathop{\mathrm{colim}}\nolimits _{(U, s)} h_ U(X)$ corresponding to $(X, t)$ and $\text{id}_ X \in h_ X(X)$.
We omit the proof of the final statement.
$\square$
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